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     \begin{table}[htbp] 
  \centering
  \caption{\bf  Estimating Time-Varying Number of Risk Factors}
\medskip
 \begin{tabular}{llllllllll}
   & $N$ & $T$ & $PC_{p1}$ & $IC_{p1}$ & $AIC_1$ & $BIC_1$ & $PC(0.10) $ & $ PC(0.15) $ & $ PC(0.20) $ \\ \hline 
    & 807    & 10     & $\begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10  \end{array}  $   & $ \begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10   \end{array}    $   &  $ \begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10    \end{array}    $   & $ \begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10   \end{array}  $     &   $ \begin{array}{c} 1 \\ 4 \\ 8 \\ 4.26 \\ 1  \end{array} $    & $ \begin{array}{c} 1 \\ 4 \\ 6.56 \\ 3.78 \\ 1  \end{array}       $    & 
    $ \begin{array}{c} 1 \\ 3 \\ 6 \\ 3 \\ 1  \end{array}    $     \\  \hline
        & 807    & 15     & $ \begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10    \end{array}   $  &  $ \begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10    \end{array} $      & $  \begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10    \end{array}   $    & $ \begin{array}{c} 10 \\ 10 \\ 10 \\10 \\ 10    \end{array}    $   &  $ \begin{array}{c} 1 \\ 4.22 \\ 8.98 \\ 5.08 \\ 1  \end{array} $    & $ \begin{array}{c} 1 \\ 4 \\  8 \\ 4  \\ 1  \end{array}    $       & $ \begin{array}{c} 1 \\ 4 \\ 7  \\ 4 \\ 1  \end{array} $        \\  \hline
    & 807    & 50     & $ \begin{array}{c} 1 \\ 4 \\ 8 \\ 4 \\ 1  \end{array}  $   & $   \begin{array}{c} 1 \\ 4 \\ 8 \\ 4 \\ 1  \end{array}    $   &  $ \begin{array}{c} 9.5 \\ 9.94 \\ 10 \\ 9.88 \\ 9.86  \end{array}  $     & $ \begin{array}{c} 3.08 \\ 5.28 \\ 9 \\ 5.2 \\ 3.1  \end{array}  $     &  $ \begin{array}{c} 1 \\ 4 \\ 8 \\ 4 \\ 1  \end{array}   $  &  $ \begin{array}{c} 1 \\ 4 \\  8 \\ 4  \\ 1  \end{array}       $ 
       & $ \begin{array}{c} 1 \\ 4 \\ 8  \\ 4 \\ 1  \end{array}        $  \\  \hline
    & 807    & 100     & $ \begin{array}{c} 1 \\ 4 \\ 8 \\ 4 \\ 1  \end{array} $    &   $ \begin{array}{c} 1 \\ 4 \\ 8 \\ 4 \\ 1  \end{array}   $    &   $ \begin{array}{c} 4.62 \\ 6.34 \\ 9.08 \\ 6.12 \\ 4.52  \end{array}  $      & $ \begin{array}{c} 1.48 \\ 4 \\ 8 \\ 4 \\ 1  \end{array}   $    &  $  \begin{array}{c} 2.04 \\ 10 \\ 10 \\ 10 \\ 2.06  \end{array}  $   & $ \begin{array}{c} 2 \\ 5 \\  10 \\ 4  \\ 1  \end{array}  $             & 
    $ \begin{array}{c} 2 \\ 5 \\ 9  \\ 5 \\ 1.38  \end{array} $        \\  \hline
\end{tabular}
    \parbox{5.75in}{\footnotesize{This table presents the estimated number of factors, $ \tilde{r}$ (average across $1,000$ Monte Carlo iterations), where I search for
 $ 0 \le r \le 10 $. The data are simulated using as parameters the values estimated with the dataset described in Section~7,  according to
$ x_{is} = {\mu}_i +  \boldsymbol{\lambda}_{i}^{r_s \prime }  {\bf f }_s^{r_s} + e_{is},  \, i=1, \cdots , 803 ,  t=1, \cdots , 500 $,
 with $  \boldsymbol{\lambda}_{i}^{r_s} \sim NID ( \boldsymbol{\mu }_{\lambda}^{r_s}, {\bf V }_\lambda^{r_s} ) $,   
  $  {\bf f }_{s}^{r_s} \sim NID ( {\bf 0 }^{r_s}, {\bf V }_f^{r_s} ) $,  and $e_{is} \sim NID(0,(\sigma_i^{r_s})^2) $,  where $ {\mu}_i  , \sigma_i^{r_s} ,  \boldsymbol{\mu }_{\lambda}^{r_s}, {\bf V }_\lambda^{r_s} ,  {\bf V }_f^{r_s} $ are
  of dimension $1\times 1, 1 \times 1, r_s \times 1 , r_s \times r_s $, and $ r_s \times r_s $, respectively, and are  estimated from the data. Here   $r_s$ indicates the (time-varying) number of risk factors corresponding to time $s$  set as $ r_s =1 , s \in [1,100], r_s =4 , s \in [101,200],
r_s =8 , s \in [201,300], r_s =4 , s \in [301,400], r_s =1 , s \in [401,500] $.
 Columns three to six correspond to the competing criteria $IP_{p1}, IC_{p1}, AIC_1, BIC_1$ (see Bai and Ng (2002)[Section 5] for details). Columns seven  to nine correspond to my large-$N$ criterion $ (T/(T-k))   V(k) +k  g(N) $  with $ g(N)  =  (30)^{-1} {   N^{\epsilon_2} \over T  \sqrt{N} } $, where $ \epsilon_2= 0.1, 0.15, 0.2 $.
   }}
\label{FIG14MCOA}
\end{table}


	
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